520 
[540 
540. 
ON A PROPERTY OF THE TOPSE CIRCUMSCRIBED ABOUT 
TWO QUADRIC SURFACES. 
[From the Messenger of Mathematics, vol. I. (1872), pp. Ill, 112.] 
The property mentioned by Mr Townsend in his paper( 1 ) in the August No., “On 
a Property in the Theory of Confocal Quadrics,” may be demonstrated in a form 
which, it appears to me, better exhibits the foundation and significance of the theorem. 
Starting with two given quadric surfaces, the torse circumscribed about these 
touches each of a singly infinite series of quadric surfaces, any two of which may be 
used (instead of the two given surfaces) to determine the torse; in the series are 
included four conics, one of them in each of the planes of the self-conjugate tetrahedron 
of the two given surfaces; and if we attend to only two of these conics, the two 
conics are in fact any two conics whatever, and the torse is the circumscribed torse 
of the two conics; or, what is the same thing, it is the envelope of the common 
tangent-planes of the two conics. 
Consider now two conics U, U', the planes of which intersect in a line I; and 
let I meet U in the points L, M, and meet U' in the points L', M': take A the 
pole of I in regard to the conic U, and A' the pole of T in regard to the conic U\ 
Take T any point on /, and draw TP touching U in P, and TP' touching V 
in P': the points P, P' may be considered as corresponding points on the two conics 
respectively. 
Join AP and produce it to meet the line I in G; the line APG is in fact the 
polar of T in regard to the conic U (for T being a point on I, the polar of T 
passes through A ; and this polar also passes through P); that is, the points T, G 
and L, M are harmonics on the line 7; whence also, in the plane of the conic U', 
t 1 Messenger of Mathematics, same volume, pp. 49, 50.]